QUANTIZATION OF SYMPLECTIC REDUCTION
MICHAEL ANSHELEVICH
Abstract. Symplectic reduction, also known as Marsden-Weinstein reduction, is an important
construction in Poisson geometry. Following N.P. Landsman [22], we propose a quantization of
this procedure by means of M. Rieffel’s theory of induced representations. Here to an equivariant
momentum map there corresponds an operator-valued rigged inner product. We define the operatoralgebraic notions that are involved in this construction, and give a number of examples.
Acknowledgements: This report is mainly based on a series of papers by N. Landsman ([19, 20,
21, 23] and especially [22]). I also would like to thank Prof. Rieffel for a helpful discussion. And of
course many ideas in this report have been touched upon in Prof. Weinstein’s course.
Added in print: (1) The contemporary name for the structure with the rigged inner product is a
Hilbert module. (2) Some related articles are: Survey of Strict Deformation Quantization by Bina
Bhattacharyya (written for the same course, http://math.berkeley.edu/~alanw/242papers.html),
and Morita Equivalence in Algebra and Geometry by Ralf Meyer, Von Neumann Algebras and Poisson Manifolds by Dimitri Shlyakhtenko, both written for Prof. Weinstein’s course Geometric Models
for Noncommutative Algebras, http://math.berkeley.edu/~aweinst/277papers/277papers.html.
1. Introduction
Symplectic reduction [1, 12, 28] (also known as Marsden-Weinstein reduction) is one of the basic
constructions in symplectic geometry. Given a Lie group G and a hamiltonian action of G on a
symplectic manifold S, one gets a momentum mapping J : S → g∗ from S to the dual of the Lie
algebra of G. If the action is strongly hamiltonian, the momentum map is G-equivariant (here G
acts on g∗ by the coadjoint action). In this case given a coadjoint orbit O ⊂ g∗ one can obtain the
reduced manifold J −1 (O)/G which has a natural symplectic structure induced from S × O (cf. the
end of Section 2).
In the standard transition from the classical to the quantum mechanical picture ([1], see also
[9] for a somewhat different approach) one replaces the symplectic manifolds S and O by Hilbert
spaces HS and HO , and actions of G on S and O by unitary representations of G on HS and
HO (more precisely, one considers projective Hilbert spaces PH, i.e. vectors are defined only up to
phase=(complex factor of modulus 1)). One usually takes the representation on HO to be irreducible.
One motivation for this is that since the coadjoint representation of G on O is transitive, the only Ginvariant functions on O, i.e. functions commuting with the action of G, are constant, cf. Example 3.
Another is that by a theorem of Kirillov [17], for G nilpotent its space of irreducible representations
is actually homeomorphic to the space of coadjoint orbits of g∗ (both with appropriate topologies).
We will consider a slightly different setting which will allow us to quantize Poisson manifolds as
well as symplectic manifolds. Namely, the definition of the Poisson manifold suggests that the main
object one works with is not the manifold itself but the Poisson algebra of functions C ∞ (P ) on it.
Correspondingly, in the quantum case it is natural to consider not the Hilbert space H, to which
corresponds the whole algebra B(H) of bounded operators, but certain subalgebras of B(H). Two
natural classes of such subalgebras are the following [3, 6, 7, 14]:
Date: September 10, 2000.
This paper was written in Spring 1996 for Prof. Weinstein’s course Symplectic Geometry at University of California,
Berkeley.
1
2
M. ANSHELEVICH
Definition 1. An operator ∗ -algebra A ⊂ B(H) is a C ∗ -algebra if it is closed in the topology induced
by the operator norm on B(H). (An algebra is a ∗ -algebra if it is closed under the adjoint operation.)
Definition 2. An operator ∗ -algebra A ⊂ B(H) is a von Neumann algebra (or a W ∗ -algebra) if it
is closed in the strong operator topology (i.e. the weak topology induced by the linear functionals
a 7→ kaξk, ∀ξ ∈ H).
Example 1. Von Neumann algebras (in particular C ∗ -algebras): B(H), in particular Mn (C), their
direct sums and tensor products; group von Neumann algebras (to be defined below).
Example 2. C ∗ -algebras that are not von Neumann algebras: algebra of compact operators.
See also Theorems 2 and 3.
The advantage of considering operator algebras other than B(H) is that once they are defined
they can be considered without reference to any particular representation on a Hilbert space (in fact
it is possible to define C ∗ - and W ∗ -algebras without reference to any particular B(H)). One usually
takes quantum algebras of observables to be C ∗ -algebras (Haag-Kastler [13], cf. also [3]), but since
the main object below is the reduced rather than the enveloping group C ∗ -algebra we might as well
restrict our attention to von Neumann algebras. While the constructions below remain valid in
the C ∗ -algebraic case, the W ∗ case is somewhat more conceptual, mainly because of the following
powerful characterization [3, 7, 14]:
Theorem 1 (von Neumann bicommutant theorem). For a ∗ -algebra A ⊂ B(H) containing the
identity of B(H), A is a von Neumann algebra iff A00 = A.
Here A0 , the commutant of A, is the algebra of all operators in B(H) commuting with all operators
in A.
Corollary and Example 3. If a von Neumann algebra A ⊂ B(H) and the representation of A on
H is irreducible, the space of the intertwining operators from the representation to itself, which is
precisely the commutant of the representation A, is reduced to the scalars, and so A = A00 = B(H).
Thus for an irreducible representation there is no difference between considering the von Neumann
algebra or the Hilbert space itself.
The difference between C ∗ - and von Neumann algebras should be clear from the following characterization of the commutative case (see e.g. [5]):
Theorem 2. A commutative C ∗ -algebra with identity is isometrically ∗ -isomorphic to the algebra
C(X) of continuous functions on some compact space X. Note that C(X) is an algebra under
pointwise multiplication.
Theorem 3. A commutative von Neumann algebra (with identity) on a separable (i.e. countablydimensional) Hilbert space is (spacially) isomorphic to the algebra L∞ (X, µ) of bounded measurable
functions on a compact metric space X, where µ is a positive Borel measure with support X.
Thus intuitively C ∗ -algebras correspond to algebras of continuous functions while von Neumann
algebras correspond to algebras of measurable functions. One of the important issues in quantization
is that we want to quantize not just “the space of functions on a manifold”, but a specific class of
them, e.g. smooth functions. By considering von Neumann algebras we avoid this issue completely,
and it is not surprising that the arguments simplify.
2. Generalized Symplectic Reduction
Thus, to get back to the construction, we quantize the classical situation
J
O ,→ g∗ ←− S
all with G actions which are assumed to be equivariant, as follows:
B(HO ) ←− Bg∗ −→ B(HS )
QUANTIZATION OF SYMPLECTIC REDUCTION
3
with G actions on all three spaces. We will suggest a plausible candidate for Bg∗ in Section 4
below. For the moment we note that given this data the symplectic reduction procedure produces
a symplectic manifold J −1 (O)/G and any function f in C ∞ (S) that is G-invariant projects down
to a function in C ∞ (J −1 (O)/G). (We will assume that the manifold S is connected and consider
the equivalent requirement that f Poisson commute with J ∗ C ∞ (g∗ )). The quantization of this data
is that for any subalgebra A ⊂ B(HS ) commuting with Bg∗ , given a representation B → B(HO )
one obtains a representation A → B(HJ −1 (O)/G ); representations of B induce representations of A.
Note that the quantum construction does not involve the G-actions or the fact that B comes from
a particular manifold g∗ .
Now taking the quantum case as a motivation and translating back to the classical situation, we
would like to have the following more general reduction procedure: given two classical representations
ρ
J
Sρ −→ P ←− S, where Sρ and S are symplectic and P is a Poisson manifold, to get a symplectic
manifold S ρ carrying a representation of the subalgebra of C ∞ (S) of functions Poisson commuting
with J ∗ C ∞ (P ). Under quite general conditions this has in fact been achieved by Xu [40, Prop. 2.1]
and, in the form below, by Landsman [22, Thm.2].
Definition 3. Let S, Sρ be two connected symplectic manifolds, P a Poisson manifold. Let
J : S → P − and ρ : Sρ → P be two classical representations, i.e. Poisson maps. Then S∗P Sρ ⊂ S × Sρ
is defined by
S∗P Sρ = {(x, y) ∈ S × Sρ |J(x) = ρ(y)}
S∗P Sρ −−−→


Thus S∗P Sρ completes the diagram y
Sρ

ρ
y Given f ∈ C ∞ (P ), we also define a vector
J
S
−−−→ P
field X̂f on S × Sρ by X̂f (g) =
− ρ∗ f, g}, using the product symplectic structure on S × Sρ .
Remark 1. Note that (ρ∗ f − J ∗ f )(x, y) = f (ρ(y)) − f (J(x)), and so S∗P Sρ is precisely the zero
manifold of all such functions.
Remark 2. If P is a linear Poisson manifold, for example a dual of a Lie algebra, then we can define
a momentum map Φ : S − × S ρ → P by Φ = ρ − J, composed with the appropriate projections. In
this case S∗P Sρ ⊂ S × Sρ is just Φ−1 (0) and X̂f is the hamiltonian vector field of Φ∗ f .
Theorem 4. S∗P Sρ is an immersed coisotropic submanifold of S × Sρ . The collection of vector
fields {X̂f |f ∈ C ∞ (P )} defines a (generally singular) foliation F of S∗P Sρ whose leaf space S ρ =
(S∗P Sρ )/F coincides with the quotient of S∗P Sρ by its characteristic foliation.
{J ∗ f
Proof. Denote M = S∗P Sρ for convenience. We show that F is, on one hand, the singular foliation
for M and, on the other hand, is tangent to it. For X ∈ Tx S and Y ∈ Ty Sρ , we have X + Y ∈ T(x,y) M
iff J∗ X = ρ∗ Y . The dimension of T(x,y) M at any point (x, y) equals (dimS+dimSρ − (rankJ∗ )(x)).
Therefore the dimension of (T(x,y) M )⊥ in T (S × Sρ ) is (rankJ∗ )(x). Let F(x,y) be the linear span
of all the vector fields {X̂f |f ∈ C ∞ (P )} at the point (x, y). Then dimF(x,y) is (rankJ∗ )(x) as well.
Therefore to show that F(x,y) = (T(x,y) M )⊥ , it suffices to show F(x,y) ⊂ (T(x,y) M )⊥ . Indeed, for
X + Y ∈ T(x,y) M as above and ω = ωS + ωSρ the symplectic form on S × Sρ , we have
ω(X + Y, X̂f )(x,y) = (d(J ∗ f − ρ∗ f )(X + Y ))(x,y) = 0
Moreover, F(x,y) ⊂ (T(x,y) M ) as well: if Xg is the hamiltonian vector field of a function g, then by
Lemma 1.2 in [38] J∗ XJ ∗ f = −Xf , where Xf is defined w.r.t. the Poisson bracket on P (rather
than P − , hence the sign), and ρ∗ Xρ∗ f = Xf . Thus X̂f = XJ ∗ f − Xρ∗ f ∈ T M . Therefore S∗P Sρ
is an immersed coisotropic submanifold in S × Sρ . Furthermore, [X̂f , X̂g ] = −X̂{f,g} (using the
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M. ANSHELEVICH
Poisson structure on P ), so by the Stefan-Sussmann theorems (cf. [25, Thm. 3.9, 3.10, App. 3])
the distribution F defines a (singular) foliation.
Remark 3. Extra regularity conditions are required to guarantee that S ρ be a manifold.
Corollary 1. Let A ⊂ (J ∗ C ∞ (P ))0 be a Poisson subalgebra of C ∞ (S). Then the map π ρ : A → C ∞ (S ρ ),
defined by
π ρ (f )(x, y) = f (x)
is well defined, and is a Poisson map. Here (x, y) is the equivalence class of (x, y) in S ρ , and prime
denotes the Poisson commutant.
θ
Remark 4. In many examples, A = θ∗ C ∞ (Q) for some classical representation S −→ Q, Q Poisson,
in which case we get a classical representation S ρ → Q. Thus given the data
θ
J
ρ
Q ←− S −→ P − , Sρ −→ P
we obtain a new representation
S ρ −→ Q
cf. Section 3 and Definition 6.
In the Marsden-Weinstein case of a group action, we have P = g∗ , a dual of a Lie algebra and
Sρ = O, a coadjoint orbit, with ρ the inclusion map. The property of P implies that we have
the product momentum map Φ : S − × O → g∗ , and the fact that the inclusion map is injective
implies that the projection of S∗P O onto J −1 (O) ⊂ S is injective. Therefore in this case we may
consider S∗P O as a submanifold of S, and the reduced manifold as its quotient. Note, however,
that while J −1 (O) is a coisotropic submanifold of S, its characteristic foliation is different from its
G-foliation, while they do coincide for S∗P O ⊂ S − × O (as in the Theorem). Thus the isomorphism
S ×∗g O/F ∼
= J −1 (O)/G is an efficient way of providing J −1 (O)/G with its correct symplectic
structure (the usual way of defining it is by the isomorphism J −1 (O)/G ∼
= J −1 (µ)/Gµ for µ ∈ O).
3. Rieffel Induction
This construction is a possible quantum counterpart of the generalized Marsden-Weinstein reduction. One is given two algebras A and B and a vector space L which is a left A- module and a right
B-module, A → End(L) ← B. A and B are usually taken to be C ∗ -algebras, but in some examples
only weaker conditions are satisfied and one has to adjust the construction accordingly. L is called
a Hilbert (C ∗ -) bimodule [18]. Given a representation πχ of B on a Hilbert space Hχ , we want to
induce a representation of A on some other Hilbert space Hχ . For example, if B is a subalgebra of A
and L = A, we want to know which representations of an algebra come from those of a subalgebra.
The corresponding problem for groups has been considered by G. Frobenius [10] and, in a more
functional-analytic context, by G. Mackey (see e.g. [26, 27] and references in [32]). M. Rieffel found
the following general construction [32]. The basic idea probably goes back to Stinespring [36].
Suppose we are given on L a rigging map, i.e. a B-valued inner product h·, ·iB . This is a linear
map L × L → B satisfying the following conditions for all ψ, φ ∈ L:
(1): hλψ, µφiB = λµhψ, φiB ∀λ, µ ∈ C (C-sesquilinearity)
(2): hψ, φiB ∗ = hφ, ψiB (B- sesquilinearity)
(3): hψ, φbiB = hψ, φiB b ∀b ∈ B (connection with the B-action)
(4): haψ, φiB = hψ, a∗ φiB ∀a ∈ A (connection with the A-action)
We will see examples of such maps below. Given a rigging map and a representation of B on
Hχ , we can form the (say, algebraic) tensor product L ⊗ Hχ . If the rigging map is positive, i.e.
∀ψ ∈ L, hψ, ψiB ≥ 0 is a positive operator in B, or at least that πχ is L-positive in the sense that
QUANTIZATION OF SYMPLECTIC REDUCTION
5
∀ψ ∈ L, πχ (hψ, ψiB ) ≥ 0 is a positive operator on Hχ , then the following bilinear form on L ⊗ Hχ
is positive semi- definite:
(ψ ⊗ v, φ ⊗ w)0 = (πχ (hφ, ψiB )v, w)Hχ
where (·, ·)Hχ is the inner product on Hχ .
Then given an A-B-bimodule L with a rigging map on it and a representation of B on Hχ , we
construct a representation of A as follows:
Step 1: Form the algebraic tensor product L ⊗ Hχ with a positive semi- definite bilinear form
on it, as above.
Step 2: Form the tensor product over B L⊗B Hχ . This is possible since L is a right B-module
and Hχ is a left B-module. The bilinear form projects to this tensor product by condition
(3) in the definition of the rigging map.
Step 3: Quotient L⊗B Hχ out by the subspace of elements of norm zero (where the norm is
induced by the above bilinear map). We obtain a vector space with a positive definite bilinear
form. After completing with respect to that form we obtain a Hilbert space Hχ . The (left)
representation of A on L carries over to the space Hχ , because of the conditions (3) and (4).
If A is a C ∗ -algebra, this representation is actually bounded. If A is not complete in the C ∗ norm, we need an extra boundedness condition on the rigging map and the representation
πχ .
We have the following analogy with Theorem 4 (see especially Remark 4). The momentum
map J : S → P − is both a map between spaces and induces a Lie algebra antihomomorphism
J ∗ : C ∞ (P ) → C ∞ (S). It corresponds both to the rigging map L × L → B and the antirepresentation End(L) ← B. Then given a representation S ρ → P , we (1) form the direct product S × S ρ
(3) take a coisotropic submanifold S∗P S ρ and (2) quotient out by the foliation generated by the
hamiltonian vector fields of the pullbacks of functions on P . The analogy is not exact. First, in
the algebraic approach the regularity questions (whether Sρ is a manifold) are missing. Second, in
the quantum case Step 2 (product over B) is not really necessary: it is implied by Step 3. This
is certainly not true in the commutative case. Third, and perhaps most important, unlike in the
commutative case, the existence of a representation by no means implies the existence of a rigging
map. In fact in specific examples (i.e. in physics) finding the rigging map is the main issue. We will
see, however, that the analogy is very close in the original Marsden-Weinstein/Mackey case, i.e. for
group actions.
We consider a sequence of examples of increasing complexity.
4. The Group Algebra
The simplest example of symplectic reduction is the following: for a Lie group G, the action of G
on itself by right translation induces a strongly hamiltonian action of G on T ∗ G, with the momentum
map J : T ∗ G → g∗− being just the fiberwise dual of the vertical inclusion maps g → T G. J ∗ C ∞ (g∗− )
are the hamiltonian functions of the left-invariant hamiltonian vector fields, i.e. the left- invariant
functions. Their Poisson commutant is precisely the right invariant functions, thus in this case
we have the dual Poisson manifold T ∗ G/G = g∗ , with the dual momentum map J − : T ∗ G → g∗
induced by the left action of G (see Definition 6). The representation of C ∞ (g∗ ) on a coadjoint orbit
O induces again the representation of C ∞ (g∗ ) on O.
We claim that the appropriate quantum analog of g∗ is the group von Neumann algebra W ∗ (G)
[5, 7, 14].
Definition and Construction 4. Let G be a locally compact (for example discrete) group. We
have a (left) Haar measure dx on G which to simplify notation, is assumed to be unimodular. We
can define the Hilbert space L2 (G, dx). G acts on itself, and hence on L2 (G), by left translation,
6
M. ANSHELEVICH
thus inducing the left regular representation g 7→ Ug of G on L2 (G). We extend this representation
to functions on G by using the following integrated form σ:
for x, y ∈ G, ξ ∈ L2 (G), f, g ∈ L1 (G)
(Uy ξ)(x) = ξ(y −1 x)
Z
Z
(σ(f )ξ)(x) = ( f (y)(Uy ξ)dy)(x) =
f (y)ξ(y −1 x)dy
G
To make this a
by
∗ -homomorphism,
G
we are forced to define the convolution multiplication on L1 (G)
Z
(f ∗ g)(x) =
f (y)g(y −1 x)dy
G
and the involution by
f ∗ (x) = f (x−1 )
Then the above representation on L2 (G) is a faithful ∗ -representation. The W ∗ - completion of
in this representation is called the group von Neumann algebra of G. One of the simplifications
of the W ∗ -approach is that the result is the same if we start with functions of compact or even finite
support on G. In particular, Ug ∈ W ∗ (G). Note also that in general a function from a set X into a
set Y can be thought of as a formal linear combination of elements of X with coefficients in Y .
We claim that W ∗ (G) is the appropriate quantization of C ∞ (g∗ ). Very superficially, we can say
that the convolution algebra of functions on G approximates the convolution algebra of functions
on g (and vice versa), which in turn is equal to the algebra of functions on g∗ under pointwise
multiplication (the Fourier transform takes convolution to multiplication and moreover takes the L1
norm induced from the left regular representation to the L∞ norm). This statement can be made
much more formal.
Among the numerous approaches to the problem of quantization one of the most common is the
(formal) deformation quantization, introduced by Bayen et al. [2]. Given a Poisson algebra A,
one considers a family of associative algebras Ah , A0 = A indexed by a real parameter h (Plank’s
constant). The underlying space of Ah is the space of formal power series in h with coefficients in
A. The multiplication maps ∗h are required to have the following properties: for a, b ∈ A
L1 (G)
a ∗h b = ab + O(h)
i
− (a ∗h b − b ∗h a) = {a, b} + O(h)
h
The construction is completely formal and does not treat the questions of convergence and topology in general. To rectify this Rieffel has introduced the concept of a strict deformation quantization
[33]. Very briefly, sometimes one can include all the algebras Ah in a single C ∗ -algebra so that the
multiplications are all induced from the ambient algebra. Then we get consistent topologies on the
whole family Ah .
The construction of the deformation quantization of g∗ (i.e. of the algebra of C ∞ functions on it)
goes as follows [34]: for a parameter h we define a new bracket on g: [X, Y ]h = h[X, Y ]. This defines
a family of Lie algebras gh with g as the underlying space; g1 = g, g0 abelian. By exponentiating we
obtain a family of simply connected Lie groups Gh ; G1 = G, G0 = g abelian. Note that the above
description of Ah is very similar to the way one obtains a Lie group from a Lie algebra. At least
locally, we can transfer the convolution multiplication from Gh to gh via the exponential map, and
then to a deformed product ∗h on g∗ via the Fourier transform scaled by h. This does in fact give a
deformation quantization of g∗ , which is even strict in the case when g is nilpotent. Of course, lots
of important details are missing in this exposition.
QUANTIZATION OF SYMPLECTIC REDUCTION
7
The above discussion was supposed to demonstrate that the appropriate quantum analog of g∗ is
the group algebra W ∗ (G). Accordingly, the quantization of the data
J−
J
g∗ ←− T ∗ G −→ g∗− , O ,→ g∗
is
W ∗ (G) −→ B(L2 (G)) ←− W ∗ (G), W ∗ (G) → B(HO )
The analog of the statement that the Poisson commutant of the space of left-invariant functions
are the right-invariant functions is that the commutant of the left regular representation is precisely
the right regular anti-representation.
We can apply Rieffel’s induction procedure to this situation. The rigging map on the dense
subspace L1 (G) ∩ L2 (G) of L2 (G) with values in L1 (G) ⊂ W ∗ (G) is just the convolution product.
We form L2 (G) ⊗ HO , with norm
kf ⊗ ηk2 = ((f, f )W ∗ (G) η, η)
HO
= ((f ∗ ∗ f )η, η) ≥ 0
The norm is equal to 0 iff f is in the kernel of the representation on HO . Thus the induced
representation we obtain is the same representation, just like in the Poisson case.
5. Crossed Products
The next simplest situation is the action on T ∗ G of a subgroup H ≤ G.
J
T ∗ G/H ←− T ∗ G −→ h∗−
J∗
C ∞ (T ∗ G)H = C ∞ (T ∗ G/H) −→ C ∞ (T ∗ G) ←− C ∞ (h∗− )
The appropriate operator algebraic notion in this case is one of a crossed (or semidirect) product
[3, 7, 14, 30]. Suppose we are given an algebra A ⊂ B(H) and a group G acting on it by an action
β. Form the Hilbert space L2 (G, H) = L2 (G) ⊗ H. We have representations π of A and U of G (by
representations of a group we mean unitary representations) on this space as follows:
x, y ∈ G, a ∈ A, ξ ∈ L2 (G, H), f, g ∈ L1 (G, A)
(Uy ξ)(x) = ξ(y −1 x)
(π(a)ξ)(x) = βx−1 (a)(ξ(x))
Thus the representation of G is just the left regular representation, but the representation of A is
twisted by β. The purpose of this is that
((Uy π(a)Uy −1 )ξ)(x) = (π(a)Uy −1 )ξ(y −1 x) = βx−1 y (a)((Uy −1 ξ)(y −1 x))
= βx−1 (βy (a))ξ(x) = (π(βy (a))ξ)(x)
Uy π(a)Uy −1 = π(βy (a))
Thus in this representation the β-action is just the conjugation action. Again we construct the
integrated form representation
Z
Z
(σ(f )ξ)(x) = ( π(f (y))Uy ξdy)(x) =
π(βx−1 (f (y))ξ(y −1 x)dy
G
G
and in order to make this a ∗ -representation have to define
Z
f (y)βy (g(y −1 x))dy
(f ∗β g)(x) =
G
∗
f (x) = βx (f (x−1 ) )
∗β
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M. ANSHELEVICH
Taking the von Neumann closure in this representation we obtain the crossed product algebra
A×β G. In the case when A is a commutative algebra, A×β G is also called the transformation group
algebra. The connection with the semidirect products is that if A = W ∗ (K) (so G acts on K by β)
and the group H is the semidirect product K×β G, then
A×β G = W ∗ (K)×β G = W ∗ (K×β G) = W ∗ (H)
Consider the crossed product C0 (G/H)×β G, where G is a locally compact group, H a subgroup,
and the action of G on the homogeneous space G/H is induced from the right action of G on itself.
For H = G, this is just W ∗ (G), which as we saw above, is a quantization of C ∞ (g∗ ). By methods
very similar to Rieffel’s approach above (which in turn has been inspired by the original quantum
mechanical Weyl quantization of T ∗ Rn using the Moyal product [9], cf. Example 4) Landsman shows
in [19] that such a crossed product is naturally a deformation quantization of the Poisson algebra
C(T ∗ G/H). It is not hard to show that the Poisson manifolds h∗ (given by the left action of H
on G) and T ∗ G/H are dual Poisson manifolds in T ∗ G. On the other hand, the representations of
W ∗ (H) (again given by the left action of H on G) and the crossed product C0 (G/H)×β G coming
from the right action of G on G on the Hilbert space L2 (G) are precisely the commutants of each
other.
6. The Universal Phase Space
Now we consider the action of a Lie group H on the cotangent bundle of an arbitrary manifold
P (P need not be Poisson). Actually, from the physical point of view it is natural to assume that
P is a principal H-bundle over some manifold Q. In this case H acts on P (and hence on T ∗ P ) in
a natural way. Note also that if P itself is a Lie group we get the situation in the previous section.
Thus the picture is
J
T ∗ P/H ←− T ∗ P −→ h∗−
We want to quantize T ∗ P/H. It follows from Landsman’s arguments in [21] (applied to the von
Neumann algebra context) that such a quantization is W ∗ (H) ⊗ B(L2 (Q)) (the action of H on H
being the right action). In fact it is easy to see that this is the commutant of the left representation
of W ∗ (H) on L2 (P ). Indeed, while P is only a vector bundle over Q, a choice of a (measurable)
cross-section gives a (non-canonical) splitting L2 (P ) = L2 (H) ⊗ L2 (Q). The representation of
W ∗ (H) on this tensor product is (left regular representation on L2 (H))⊗(identity on L2 (Q). By an
important (and very non- trivial) theorem [16, Ch. 13], a commutant of a tensor product is a tensor
product of commutants. Therefore the commutant in this case is (right regular representation on
L2 (H))⊗B(L2 (Q)), as stated.
Remark 5. This fact can also be shown by elementary means using the Hilbert-Schmidt integral
operators in L2 (P × P ) [21, Sec.3].
The group action of a group H on a cotangent bundle of a principal H-bundle P with base space
Q has a very natural physical meaning. It was first pointed out by Sternberg [35] and refined in
[29, 37]. For H = U (1), the reduction at a coadjoint orbit O = {e} = (point) corresponds to the
description of the motion of a particle with charge {e} in the electromagnetic field. More generally,
for arbitrary H one thinks of the interaction of the Yang-Mills field with the gauge group H with
a particle with “charge” = coadjoint orbit of H. An important point here is that while P , and
hence T ∗ P and the reduced manifold J −1 (O)/H are bundles over Q, they are not naturally bundles
over T ∗ Q. However, J −1 (O)/H and hence T ∗ P do become bundles over T ∗ Q given a choice of a
connection on P .
As stated above, the main obstacle to applying Rieffel’s induction procedure is finding the
operator-valued inner product on the given bimodule. In the classical situation, while one does
not always obtain a dual pair of Poisson manifolds given just a map S → P , one has the reduction
QUANTIZATION OF SYMPLECTIC REDUCTION
9
procedure in the case where P = g∗ , under mild conditions on the action, by the standard MarsdenWeinstein reduction. We now show that in the operator algebraic case if the algebra one is inducing
from is a group algebra of a locally compact group, then the induction procedure goes through [32].
The main point is that given a representation π of W ∗ (G) on a (pre-)Hilbert space (rather than
just a vector space) H, one can define a W ∗ (G)-valued inner product on H. Indeed for ψ, φ ∈ H
one defines hψ, φiW ∗ (G) to be a function on G as follows: g 7→ (π(g)φ, ψ)H . Such a function is in
L1 (G) and therefore is an element of W ∗ (G). One also shows that this rigging map is positive if G
is compact, and L-positive (i.e. the induced inner product on the tensor product is positive) if the
representation one is inducing from is weakly contained in the left regular representation of G (all
representations of G have this property precisely when G is amenable).
7. Projective Representations
In this section we consider the generalization of Marsden-Weinstein reduction in a somewhat
different direction. Namely, we assume that the action of the group G on the symplectic manifold
S is hamiltonian but not necessarily strongly hamiltonian. That is, we still obtain the momentum
map J : S → g∗− and the corresponding map J ∗− : g → C ∞ (S − ), but the latter map is no longer
required to be a Lie algebra homomorphism. It is well known how to deal with this situation in the
classical case ([12, 28]). If {J ∗ (u), J ∗ (v)} =
6 J ∗ ([u, v]), u, v ∈ g, then the obstruction to the action
being strongly hamiltonian is given by the 2-cocycle Σ ∈ g∗ ⊗ g∗ ,
Σ(v, u) = {J ∗ (u)(s), J ∗ (v)(s)} − J ∗ ([u, v])(s)
= {hJ(s), ui, hJ(s), vi} − hJ(s), [u, v]i
which depends on the connected component of s ∈ S and not on s itself. This cocycle can be
eliminated by changing the momentum map if and only if it is a coboundary. However, in general
we can define an affine Poisson bracket {·, ·}Σ on C ∞ (g∗ ) by
]
{ũ, ṽ}Σ = [u,
v] + Σ(u, v)1g∗
where ũ ∈ C ∞ (g∗ ) is defined by ũ(θ) = hθ, ui, and 1g∗ is the identity function on g∗ . Note that the
bracket is thus uniquely defined. Then J is a Poisson map with respect to this modified Poisson
structure on g∗ , and is equivariant with respect to the original action of G on S and the modified
Σ of G on g∗ given by
coadjoint action πco
Σ
πco
(g)θ = πco (g)(θ + J(s)) − J(sg −1 )
where πco is the coadjoint action. This is again independent of s if S is connected.
Therefore in order to quantize this situation we want to quantize g∗ with affine Poisson structure
given by a 2-cocycle [22]. The appropriate operator-algebraic notion is one of a twisted group algebra.
First let the group G be discrete. In this case the group algebra of G is essentially generated by
the unitaries Ux , x ∈ G subject to the relations Ux Uy = Uxy , Ux−1 = Ux −1 , U1 = 1. In a twisted
algebra we want to modify the multiplicative relation to Ux Uy = c(x, y)Uxy . The product should be
a unitary, and the multiplication should be associative, therefore we have the following conditions
on c:
|c(x, y)| = 1
c(x, y)c(xy, z) = c(x, yz)c(y, z)
Such a c is called a multiplier (no relation to the multiplier algebra). In the quantization we
would expect the infinitesimal version of c, i.e. the 2-cocycle on g derived from it, to be −Σ (and
vice versa).
10
M. ANSHELEVICH
For such c the representation U is called a projective representation, because by a theorem of
Wigner any unitary representation on the projective Hilbert space (i.e. all the maps in the image
of the representation preserve the absolute values of inner products) lifts to unitary or antiunitary
twisted representation of the above form on the Hilbert space itself.
We proceed as in Sections 4 and 5, and use the same notation. We have a projective unitary left
regular representation of G on L2 (G)
(Uy ξ)(x) = c(x−1 , y)ξ(y −1 x)
its integrated form
Z
Z
(σ(f )ξ)(x) = (
f (y)(Uy ξ)dy)(x) =
G
f (y)c(x−1 , y)ξ(y −1 x)
G
and the twisted multiplication and involution on L1 (G)
Z
(f ∗c g)(x) =
f (y)g(y −1 x)c(y, y −1 x)
G
(f ∗c )(x) = c(x, x−1 )f (x−1 )
The twisted group algebra W ∗ (G, c) is the completion of this representation on L2 (G). We have the
rigged inner product given by the procedure at the end of Section 6.
Example 4 (Weyl Quantization). The oldest example of quantization procedure is Weyl’s quantization of T ∗ Rn . What follows is a somewhat modernized treatment of that construction. See [9] for
more explicit approach and [15] for operator-theoretic background; cf. also [31]. For simplicity we
put n = 1. Thus we consider the symplectic manifold T ∗ R = R2 with R2 acting on it by translations.
The action is not strongly hamiltonian: J ∗ (x, y)(α, β) = xα − yβ, where (x, y) lies in the Lie algebra
of R2 and (α, β) ∈ T ∗ R. Then while the regular Lie algebra of R2 is abelian, the new Lie algebra
structure on it is given essentially by the symplectic form on T ∗ R, i.e. by the cocycle
Σ((x1 , y1 ), (x2 , y2 )) = {J ∗ (x1 , y1 ), J ∗ (x2 , y2 )} = x2 y1 − x1 y2
∗
In other words, this C ∞ (R2 ) is generated by two elements x, p subject to the relations
[x, x] = 0 = [p, p]
[p, x] = 1
These relations are called canonical commutation relations (CCRs). The corresponding multiplier, obtained by “exponentiating” the above relations, is given by c((a1 x + b1 p), (a2 x + b2 p)) =
exp(i(a1 b2 − a2 b1 )/2). Thus in this case the twisted group algebra is basically generated by the
unitaries Us , s ∈ R2 subject to the relations Us Ut = exp(i[s, t]/2)Us+t , or equivalently Us Ut =
exp(i[s, t])Ut Us , where [·, ·] is now the symplectic form on R2 . These are the Weyl commutation
relations. The twisted C ∗ -algebra given by these relations is the algebra of compact operators
K(L2 (G)), and therefore the twisted von Neumann algebra is B(L2 (G)).
8. Odds and Ends
(1) One of the main results of Rieffel’s theory of induced representations is the following
Theorem 5 (Imprimitivity Theorem). [32] There exists an A-C imprimitivity bimodule L if
and only if there is a bijective correspondence between the sets of L-positive representations
of A and C. In this case A and C are called strongly Morita equivalent.
Here
QUANTIZATION OF SYMPLECTIC REDUCTION
11
Definition 5. L is an A-C imprimitivity bimodule if it has both A-valued and C- valued
rigged inner products, which satisfy the following extra compatibility conditions:
i) span{hψ, φiC |ψ, φ ∈ L} is dense in C, and similarly for A.
ii) A hψ, φiζ = ψhφ, ζiC for all ψ, φ, ζ ∈ L.
The point is, of course, that using this data we can induce representations of A from
those of C and vice versa. Note also that given just the C-valued inner product on L, we
can always find an algebra A such that the above conditions are satisfied. Indeed, given any
operator- valued inner product h·, ·i on L, we always have the second one given by “rank one
operators”: hhψ, φii : ζ → ψhφ, ζi (see [31] for the simplest case of this construction; cf. also
[9, Prop. 1.46]). It will then follow from the properties of the first rigged inner product that
the second one is again a rigged inner product, and moreover that their images commute.
Note, however, that it is not true in general that the algebra generated by these rank- one
operators (called the imprimitivity algebra of C) is the commutant of C, or indeed a von
Neumann algebra.
The classical counterparts of these notions are the following:
Definition 6. [24, 38, 41] A classical equivalence bimodule of a pair of Poisson manifolds
(P1 , P2 ) consists of a symplectic manifold S and a pair of Poisson morphisms J1 : S → P1 −
J
J
2
1
and J2 : S → P2 , such that P2 ←−
S −→
P1 − is a complete full dual pair with connected
and simply connected fibers. This means that J1 ∗ C ∞ (P1 − ) and J2 ∗ C ∞ (P2 ) are each other’s
Poisson commutants in C ∞ (S), that the leaf spaces of the foliations defined by the fibers of
J1 and J2 are manifolds in the quotient topology, and that J1 and J2 are surjective, as well
as complete Poisson maps.
In this case P1 and P2 are called Morita equivalent.
Theorem 6. [22, 39, 40, 41] Let P1 and P2 be Morita equivalent Poisson manifolds. Then
there is a bijective correspondence between their respective symplectic realizations.
Remark 6. To be more precise, the above theorem of Xu corresponds to the case of Rieffel’s
theorem where the representations are required to be strictly positive, and the above theorem
of Rieffel corresponds to the case of Xu’s theorem where the foliations of J1 and J2 are not
required to be manifolds.
(2) In a manner analogous to a group algebra one can introduce an algebra of a groupoid [4,
I.1,II.5,V.4]. Indeed, the definition of convolution of two functions on a group G can be
written as
Z
Z
−1
(f ∗ g)(x) =
f (y)g(y x) =
f (y)g(z)
y∈G
yz=x
In particular the definition uses only the groupoid property of G, i.e. the products need
not be defined for all y, z ∈ G. Under certain discreteness/smoothness conditions on the
groupoid one can define the corresponding algebra in an analogous manner. In fact, from the
point of view of operator algebras groupoid algebras are more natural, since, for example,
under quite general conditions a group transformation von Neumann algebra is determined
just by the orbit structure and does not depend on the group action itself. Moreover, the
class of groupoid von Neumann algebras is very wide [8]. It includes the group algebras, but
it also includes, for example, the matrix algebras. The groupoid giving Mn (C) is just the
pair groupoid on the set of n elements.
In [21] Landsman shows that for the Lie algebroid L(Γ) associated to a Lie groupoid Γ,
the quantization of the Poisson algebra C ∞ (L(Γ)∗ ) is the groupoid C ∗ -algebra C ∗ (Γ). Note
that this includes quantizations of C ∞ (g∗ ) as well as C ∞ (T ∗ M ). Also, the construction of
rigged inner product at the end of Section 6 works perfectly well for groupoids [22].
12
M. ANSHELEVICH
(3) Section 5 (Crossed Products) is connected with the simplest case of the basic construction of
Jones [11]. For a group G and a (say, closed and open) subgroup H, we have the left representation of W ∗ (H) on L2 (G) as the commutant of the (right) crossed product C(G/H) ×β G.
Note that L2 (G) is W ∗ (G) in any faithful GNS representation. The corresponding first step
in the Jones tower is W ∗ (H) ⊂ W ∗ (G) ⊂ C(G/H) ×β G, i.e. C(G/H) ×β G is generated by
W ∗ (G) and the orthogonal projection e1 : L2 (G) → L2 (H) as operators on L2 (G). Note
that the corresponding conditional expectation E1 : W ∗ (G) → W ∗ (H) is precisely what gives
the W ∗ (H)-valued rigged inner product on L2 (G). Indeed, this conditional expectation is
E1 : f 7→ f |H, for f ∈ L1 (G), and the rigged inner product is given by hf, gi = E1 (g ∗ ∗ f ).
Note that one of the purposes of Rieffel’s construction is to generalize the above situation
to, say, subgroups that are closed but not open.
(4) In a sense continuing the previous remark, we can generalize Section 4 (Group Algebra) to
arbitrary symplectic groupoids. On the classical side, we use the notation of Theorem 4.
α
β−
For a symplectic groupoid S with base P , we have two Poisson maps S −→ P ←− S − , so in
this case Sρ = S. The manifold S ∗P S is just the set of multipliable pairs S ×αβ S. There
is a natural map from this manifold to S given by the multiplication map.
We show that the foliation given by this map is the characteristic foliation of S ∗P S.
For this we have to use the stronger property of a symplectic groupoid that the graph of
the multiplication map Γm = {(g, h, gh)} is a lagrangian submanifold in S − × S − × S. For
convenience denote the factors by S1 , S2 , S3 , with projections π1 , π2 , π3 . Note that π1 × π2 is
an injective map of Γm onto S1 ∗P S2 , while the fibers of π3 are precisely the fibers of m. A
vector tangent to Γm can be written as X + Y , X ∈ T (S1 ∗P S2 ), Y ∈ T S3 . Now for any Ẑ as
in Theorem 4, Ẑ is in the characteristic foliation of S1 ∗P S2 , hence X ⊥ Ẑ (symplectically).
Also clearly Y ⊥ Ẑ. Therefore Ẑ ⊥ T Γm . But Γm is lagrangian, hence Ẑ ∈ T Γm . Being
also in the fiber of π3 , Ẑ has to be tangent to the multiplication foliation.
We also have to show that the reduced manifold S obtained in this way has the correct
symplectic structure.
Then taking for A the commutant of α∗ C ∞ (P ) ⊂ C ∞ (S), which is precisely β ∗ C ∞ (P ),
the induced representation will be the same representation on S, just like in the case of T ∗ G.
In the quantum case, the properties of the maps α, β of a symplectic groupoid correspond
to two representations of the same algebra A on some L which are each other’s commutants.
Moreover, the definition of a symplectic groupoid here apparently corresponds to the requirement that L be a Hilbert algebra (or possibly a left Hilbert algebra). That is, if L is a
Hilbert space it is the completion of A in some GNS representation with respect to a faithful
state (or weight), while in general vector space case we just take L = A (cf. the group case
in the previous remark). Note that this would seem to indicate that for any (quantizable)
Poisson manifold there is a symplectic groupoid having it as base. In any case, the rigged
inner product here is just the multiplication in L (or A), and the induced representation is
the same as the inducing one.
One of the most general cases when there is known a natural rigged inner product seems
to be when L is an algebra, A ⊂ L a subalgebra, and there is a conditional expectation
E : L → A. Then the construction is like in the previous comment, and again we can induce
representations of the commutant. In this context, it would be interesting to see if there is
a useful classical construction generalizing T ∗ G/H ← T ∗ G → h∗− .
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